Weighted population density scales each subdivision to have a weight proportional to its population. Mathematically, WPD = Sum(Population Density * Proportion of state's population). It's also equivalent to the density experienced by the average inhabitant of each state.

The US Census Bureau has weighted population density data for metro areas, but no one seems to have calculated such densities for states. So I pulled the census tract data from the 2010 Census and calculated in people per square mile.

New York, as expected, is a factor of four above any other state, but second place was a bit surprising. The rugged terrain of Hawaii forces settlements to build up and denser rather than spreading out, and its small size means little rural area to pull the number down. A lack of significant rural population is also why Alaska is 29th in weighted density, as Anchorage alone is nearly half the state.